K. Kumar 1, V. Singh 2 and S. Sharma 1 ABSTRACT NOMENCLATURE

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1 Journal of Applied Fluid Mechanics Vol. 0 No. pp Available online at ISSN EISSN DOI: /acadpub.jafm Effect of Horizontal Magnetic Field Horizontal Rotation on Thermosolutal Stability of a Dusty Couple-Stress Fluid through a Porous Medium: a Brinkman Model K. Kumar V. Singh S. Sharma Department of Mathematics Statistics Gurukula Kangri Vishwavidyalaya Haridwar India. Department of Applied Sciences Moradabad Institute of Technology Moradabad 4400 India. Corresponding Author kkchaudhary000@gmail.com (Received August 5 06; accepted November 7 06) ABSTRACT The problem of the onset of double diffusive convection in a couple-stress fluid saturated with a porous medium is studied under the effects of magnetic field rotation suspended dust particles. Linear stability analysis based on the method of perturbations of infinitesimal amplitude is performed using the normal mode technique for the case of free-free boundaries. The governing hydrodynamic hydromagnetic equations of fluid flow are governed by the Brinkman model. The stability analysis examines the effects of various embedded parameters for the stationary mode both analytically graphically. The principle of exchange of stabilities holds good in the absence of solute gradient parameter. Also the sufficient conditions responsible for the existence or non-existence of overstability are obtained. Keywords: Couple-stress fluid; Magnetic field; Rotation; Dust particles; Brinkman porous medium. NOMENCLATURE t time coordinate d thickness of porous fluid layer q fluid velocity vector having components qd suspended particle velocity having components p pressure T0 reference temperature S 0 reference concentration T temperature S concentration K stokes drag coefficient N 0 number density of dust particles k Darcy-brinkman medium permeability kt coefficient of heat conduction k S coefficient of solute concentration D differential operator m mass of suspended particles n frequency of the harmonic disturbance Xi gravitational acceleration vector cs heat capacity of solid material cv w s w H * specific heat of the fluid at constant volume vertical fluid velocity vertical particle velocity complex conjugate of horizontal magnetic field having components h perturbation in magnetic field strength X vertical component of current density after applying normal mode method W K Z k x k y k vertical component of fluid velocity after applying normal mode method vertical component of magnetic field after applying normal mode method vertical component of vorticity after applying normal mode method wave number in x direction wave number in y direction Resultant wave number

2 K. Kumar et al. / JAFM Vol. 0 No. pp p pressure gradient term 0 density of fluid s density of solid material fluid viscosity couple-stress fluid viscosity ef effective viscosity ef effective kinematic viscosity e magnetic permeability T thermal expansion coefficient S solute expansion coefficient T adverse temperature gradient S solute concentration gradient electrical resistivity Θ temperature component after applying normal mode method solute component after applying normal mode method p perturbation in fluid pressure p perturbation in fluid density kinematic viscosity kinematic viscoelasticity T thermal diffusivity S solute diffusivity z-component of vorticity Ω i z-component of current density horizontal rotational vector having components 3-dimensional Laplacian operator growth rate of harmonic disturbance after applying normal mode method perturbation in temperature T perturbation in solute concentration S Suspended particles radius vertical unit vector Non-dimensional Parameters a dimensionless wave number Pl dimensionless medium permeability p thermal Prtl number p magnetic Prtl number q Schmidt number B suspended particle parameter Q modified Chrasekhar s number D modified Darcy-Brinkman number R S T A A modified Darcy-brinkman thermal Rayleigh number modified solute Rayleigh number modified Taylor s number Darcy-Brinkman medium porosity modified couple-stress parameter. INTRODUCTION The study of fluid dynamics is of great importance for scientific researchers engineers to underst various significant fascinating applications of fluid mechanical phenomena such as calculating forces moments on aircrafts determining the rate of mass flow in oil industry through pipelines forecasting weather patterns understing nebulae in interstellar space in traffic engineering by considering traffic as a continuously distributed fluid. Copious literatures (Lin 955; Batchelor 967; Rajagopal 978; Drazin Reid 98; Bansal 004; Gupta Gupta 03) are available that provide a basic theoretical experimental support for various fluid dynamical phenomena as well as hydrodynamic stability of Newtonian non-newtonian fluids. The influence of magnetic field rotation on thermal instability of an incompressible Newtonian fluid layer has been the subject matter of great interest over the years since the pioneering work by Chrasekhar (98) within the framework of linear stability theory. In double diffusive (when the fluid layer is simultaneously heated soluted from underside) phenomena convection process is induced by the combined effects of temperature difference concentration difference which have different diffusion rates. Double diffusive (thermosolutal) convective phenomena through a porous medium frequently occur in seawater flow mantle flow in earth s crust ground water hydrology (underground disposal of nuclear wastes) astrophysics (helium acts like a salt in raising the density in stellar case) oceanography soil science (transportation of soil solid waste mud particles into the rivers lakes) solar system (where heat helium diffuse at differing rates) atmospheric science limnology. The presence of waste matter in water or air is responsible for certain chemical reactions helpful to underst several mass transport processes such as groundwater pollution transportation of nuclear wastes food processing industry contaminated air transport in atmosphere (presence of dust particles in atmosphere) manufacturing of glassware in polymer production. Nearly all fluid flow mechanisms happening in the universe involve circulation or rotational effects which have applications to a greater or lesser extent in a number of processes which include large scale circulations in the atmosphere oceans motion of a hurricane tornado tsunami in many small scale flows like stirring of tea in a cup. Thermo-convective phenomenon in a rotating system is of practical significance finds its applications in rotating machinery crystal growth food processing industry centrifugal casting of metals in thermal power plants (to generate electricity by the rotation of turbine blades). Greenspan (969) studied the theory 68

3 K. Kumar et al. / JAFM Vol. 0 No. pp. x-x 07. of rotating fluids which has applications in various technological situations which are governed by the action of coriolis force. The double diffusive convective problems of couple-stress fluid through an anisotropic porous medium considering rotational effect have been discussed in the references by Malashetty Kollur (0) Malashetty et al. (0). A major part of the universe is filled with charged particles magnetic field is present in around the heavenly bodies. Magnetic field plays a dominant role in several clinical purposes such as in neurology orthopaedics for examining the internal organs of the body in various diseases like tumours detection heart brain diseases stroke damage etc. The theory of magneto-hydrodynamics (MHD) has several scientific practical applications in geophysics (in the study of earth s core) atmospheric science (solar wind is governed by MHD) astrophysics plasma physics space sciences etc. Thermal instability problem of an electrically conducting couple-stress fluid heated from below through a porous medium in the presence of a uniform magnetic field has been investigated by Sharma Thakur (000). Sharma Sharma (004) have considered the effect of suspended particles on couple-stress fluid heated from below in the presence of vertical rotation vertical magnetic field noted that the effect of rotation is to stabilize the system whereas suspended particles have destabilizing effects. Thermosolutal convective problem for a couple-stress fluid through a porous medium under the influences of uniform vertical magnetic field uniform rotation has been studied by Singh Kumar (0) whereas the effect of suspended particles on couple-stress fluid heated soluted from below through a porous layer has been noted by Sunil et al. (004). Couple-stress theory due to Stokes (966) is of vital importance to underst different aspects including the lubrication mechanism functioning of synovial joints during human locomotion opened new vistas in several areas of scientific technical research. Shivakumara et al. (0) illustrated the linear nonlinear stability problem of double diffusive convective phenomena for couple-stress fluid through a porous layer. Kumar et al. (05a b 06) considered theoretically thermal convection problems for both couple-stress fluid ferrofluid to include the effects due to magnetic field rotation compressibility variable gravity heat source strength through Darcy as well as Brinkman porous medium. Thermal as well as thermosolutal convective instability problems through a porous medium have extensive attention over the years also recognized as the problems of fundamental importance in solidification chemical processing industry bio-medical sciences geophysical fluid dynamics soil sciences petroleum industry filtering technology. Extensive reviews related to thermal thermosolutal convective problems through a porous medium have been covered in the books by Ingham Pop (005) Vafai ( ) Nield Bejan (006) Vadasz (008). McDonnel (978) has conducted extensive systematic studies regarding the impact of porosity in astrophysical situations because a greater part of the universe is filled with fine dust particles. It is well known that the Darcy equation fails to give satisfactory results for high permeability porous medium. So the consideration of non-darcy (Brinkman) model which take care of boundary inertia effects is of practical interest for high permeability porous medium. Kuznetsov Nield (00) analyzed the thermal instability problem in a porous layer saturated by a nanofluid employing the Brinkman model for the porous medium. Mahajan Sharma (0) studied the asymptotic stability also determine the stability bounds for both equilibrium arbitrary flows of a couple-stress fluid in a Brinkman flow. Recently the linear nonlinear stability analysis of double-diffusive reaction convection in an anisotropic Darcy- Brinkman porous layer is performed by Gaikwad Dhanraj (06). Therefore the Brinkman model which is physically more realistic than the other ones is considered for the porous medium to include effects due to horizontal magnetic field horizontal rotation in a dusty couple-stress double-diffusive convection. Some previous published works by Sharma Thakur (000) Sharma Sharma (004) Sunil et al. (004) Singh Kumar (0) can be recovered from the present investigation.. MATHEMATICAL FORMULATION Consider an infinite horizontal porous layer of thickness d of an incompressible couple-stress fluid with dust particles heated soluted from below through a porous medium of porosity permeability k. The fluid layer is acted upon by a uniform horizontal H H 00 a uniform magnetic field horizontal rotation Ω 00. The fluid layer is heated soluted from below such that an adverse temperature gradient dt T dz concentration gradient ds S are dz maintained. The coriolis effect has been taken into account by including the coriolis force term q in the momentum equation whereas the Centrifugal force term grad r can be realized as a gradient of a scalar therefore has been absorbed into the pressure term p pf r. The term p f sts for the fluid pressure denotes the angular velocity of rotation. The Brinkman model is employed for the porous medium the fluid density variation is based on Boussinesq approximation (903). The boundaries are taken to be free maintained at uniform temperatures the porous layer 683

4 K. Kumar et al. / JAFM Vol. 0 No. pp. x-x 07. Fig. a. Geometrical Sketch of the Physical Problem. is extended infinitely in x y directions the z axis is taken vertically upward with the origin at the lower boundary. The pressure p density viscosity viscoelasticity depend upon the vertical co-ordinate z- only. The basic equations in a double diffusive convection for an incompressible couple-stress fluid saturating a Brinkman (947a b) porous medium under Boussinesq approximation are defined as The density equation of state is given by 0 T T0 T S S S 0 () Equations of momentum mass conservation are defined as p 0Xi q k 0 q 0 0. ef KN qq t q Ω q e qd q HH 4 (). q 0 (3) Neglecting the variations in pressure force Darcian force magnetic field buoyancy force (due to gravity) on the particles the equations of motion continuity for the particles are as mn q t q. q K N q q (4) d 0 d d 0 d N 0. N 0 t 0q d (5) Assuming that the suspended particles the fluid particles are in thermal solute equilibrium the equations for temperature solute concentration can be presented as T 0cv scs 0cv q. T t (6) mn 0c pt qd. T kt T t S 0cv scs 0cv q. S t mn 0cpt qd. S ks S t (7) where cv cs cpt ks denote the analogous solute coefficients. The Maxwell s (866) electromagnetic equations yield H qh H t (8). H 0 (9) 3. PERTURBATION TECHNIQUE AND LINEAR STABILITY ANALYSIS Here the stability of the basic state of system has been examined by using the perturbation technique. Let the perturbations in the basic variables such as q 000 particle velocity fluid velocity 000 q temperature T pressure p density d particle number density N 0 magnetic field H 684

5 K. Kumar et al. / JAFM Vol. 0 No. pp. x-x 07. be denoted by quvw N x y z h h h h respectively. qd l r s p After linearizing the system of Eqs. () (9) following Boussinesq approximation the perturbation equations obtained after eliminating the pressure gradient term are as follows T S 0 (0) gt S x y x y ζ ef 4 w x w t mn 0 e H w m t K t hz w x k () w ef ζ x ζ mn 0 ζ () t m t 0 K t e H ξ ζ 4 0 x k h z w H h z (3) t x ξ ζ t x H ξ (4) t E b b where t T Eb b S T S w s (5) w s (6) scs scs E E 0cv 0cv mn 0cpt mn 0cpt b b. 0cv 0cv Differential Eqs. () (6) can be solved using the method of normal modes. Suppose that the perturbations in various physical quantities have a solution with a dependence on x y t of the form z z W z Z z w ζ h z ξ K z X z exp ik x x ik y y nt (7) Using expression (7) the Eq. () (6) into nondimensional form reduce to (after dropping the asterisk for convenience) f D A D a D a Pl Pl ga d Ωikx d D a Wz Tz Sz ehik xd Zz D a Kz (8) f DA D a D a P P l l Ωik xd ehik xd Zz Wz Xz 40 ik x d H p D a (9) Kz Wz (0) ik x d H p D a T d B D a pe z T Xz Zz () Wz () S d B D a qe z Wz (3) S where the following non-dimensional quantities with the scaling have been used in Eqs. (8) (3) z a nd k z k Pl p d d d T m 0f q p N 0 B b S d K m ef P E E b E E b D l. A d The boundary conditions appropriate for the case of two free boundaries are defined as W D W X DZ 0 at z 0. DX K 0 on a perfect conducting boundary (4) Eliminating z z Zz Xz Kzfrom Eqs. (8) (3) considering an appropriate solution for W (vertical component of fluid velocity) of the form W W 0 sin l z (5) a dispersion relation is obtained as follows TA A A3A4 xcos 3 4 B i AAAA x AA 4Qx cos i QA Ax 3 xcos xa4sa RA 3 0 (6) 685

6 K. Kumar et al. / JAFM Vol. 0 No. pp. x-x 07. where R a P l x R x i 4 4 P l k l l l l d Q T A S Q T A 4 4 S l l l l l DA DA k cos x k A x ipe l i f DA A x x i P P A3 xiqe A4 xip gt Td R T number) S 4 (Darcy-Brinkman thermal Rayleigh 4 gssd (Solute Rayleigh number) S 4 4 d T A Ω (Taylor s number) e d Q H (Chrasekhar s number) 4 0 Equation (6) is the required dispersion relation accounting the effects of suspended particles horizontal magnetic field horizontal rotation medium permeability medium porosity on thermosolutal instability of a couple-stress fluid saturating a Brinkman porous medium. 4. THE STATIONARY STATE In a stationary convection the marginal state occurs when 0 (i.e. growth rate vanishes). Substituting 0 in Eq. (6) the stationary Rayleigh number R in terms of various parameters the square of is obtained as x DA Qx xcos x P R SxB Bx TA x x cos x DA x Qx cos P (7) The effect of various parameters on thermosolutal convection can be realized analytically by evaluating dr the following derivatives dr dr dr. dta ds d db dr dr dr dr dq d dd dp A Equation (7) yields dr x Pcos dt A B x G QPx cos dr (8) ds (9) A xg Q T P x G x QPx cos dr cos d B TA P x xgqpx cos dr db TA x x cos dr dq dr d (30) x G Qx x cos P B x x G Qx cos P (3) cos xg QPx cos x cos TA P x x B cos QPx cos 3 x TA P x x BPx x G 3 dr x TA P x x cos dd A BP x x G QPx cos dr x G TA x xgcos dp Bx xg QPx cos P D A. where G x (3) (33) (34) (35) From the derivatives (8) - (35) it is clear that the Taylor s number solute gradient rule out the possibility of the onset of convection whereas suspended particles medium porosity accelerate the onset of convection. The magnetic field couple-stress Darcy-Brinkman parameter have stabilizing (or destabilizing) effect the medium permeability has a destabilizing (or stabilizing) on thermosolutal instability if xg QPx cos orta Pxxcos respectively. In the absence of rotation iet.. 0 magnetic field couple-stress A Darcy-Brinkman parameters always postpone the onset of convection whereas medium permeability assures the destabilizing effect on the system. The variation in Rayleigh number R for the stationary 686

7 K. Kumar et al. / JAFM Vol. 0 No. pp. x-x 07. state against various values of parameters such as rotation solute gradient medium porosity suspended particles magnetic field couple-stress Brinkman number medium permeability on double-diffusive convection saturating a Brinkman porous medium are depicted graphically in Figs. - 8 respectively for 45. Rayleigh number R porosity= porosity=4 porosity=6 porosity=8 porosity= Rayleigh number R TA=000 TA=5000 TA=0000 TA=0000 TA= Fig. 3. Variations of R with x for various values of medium porosity 4680 B 5 P3 S 00 Q 00 D A 0 0 T A Fig.. Variations of R with x for various values of rotation parameter TA B 5 S00 P 3 D A 0 0 Q Rayleigh number R B=5 B=0 B=5 B=0 B= Rayleigh number R S=00 S=00 S=300 S=400 S= Fig. 4. Variations of R with x for various values of suspended particles B P0 5 D 5 5 Q 500 T 0 A A 000 S Fig.. Variations of R with x for various values of solute gradient parameter S B 0 P0 5 D 5 5 Q 00 T A A Rayleigh num ber R Q=0 Q=40 Q=60 Q=80 Q=00 5. PRINCIPLE OF EXCHANGE OF STABILITIES Here the conditions have been derived if any under which principle of exchange of stabilities (PES) holds true also the possibility of oscillatory modes Fig. 5. Variations of R with x for various values of magnetic field Q B 0 P 0 D 0 0 T 0 A A 000 S

8 K. Kumar et al. / JAFM Vol. 0 No. pp. x-x 07. Rayleigh number R Fig. 6. Variations of R with x for various values Rayleigh number R of couple-stress parameter 3579 B 5 5 P 5 D 0 T S 500 A A Q Fig. 7. Variations of Rwith x for various values of Darcy-Brinkman parameter DA 4860 B 3 P 5 5 Rayleigh number R couple-stress=3 couple-stress=5 couple-stress=7 couple-stress=9 couple-stress= Brinkman parameter=4 Brinkman parameter=8 Brinkman parameter= Brinkman parameter=6 Brinkman parameter=0 T 0000 S 500 Q A P=0 P=0 P=30 P=40 P=50 Fig. 8. Variations of R with x for various values of permeability P B 0 D 0 5 Q 40 A A T 0000 S For this purpose multiplying Eq. (8) by W then integrated over the range of z. Using Eqs. (9)- (3) with the help of boundary conditions (4) gives f I Pl DA gt a I Pl T p * I3 pei 4 B gs a * I5 qei6 S q B f DA d I7 I 8 Pl Pl e * e d pi 9 I0 pi I0 40p 40p (36) Substituting r ii in Eq. (36) equating real imaginary parts leads to f r g 7 T a I d I T p r i rb r i pei 4 gs a r B r i S q rb r i qei 6 e B 4 r i 0 I9 d I Id I 7 f i Id I 7 P l r i DA I I8 gt a Pl T p rb r i I3 i BpEI4 B r i 0 rb r i I5 i BqEI6 B r i gsa e d I I0 Sq 40p (37) 688

9 K. Kumar et al. / JAFM Vol. 0 No. pp. x-x 07. f g Ta I T p r i rb ri pei 4B I3 r pei 4 B r i i 0 rb ri qe I 6 gs a B I5rqE I 6 Sq B r i d f r e I9d I I7 4 r i 0 (38) where the positive defined integrals I I are defined below as I DW a W dz 0 4 I 0 D W a W a DW dz I3 0 D a dz I4 0 dz I5 0 D a dz I6 0 dz I7 0 Z dz I8 0 DZ a Z dz I9 0 DK a K dz 4 I0 D K a K a DK dz 0 I 0 X dz I DX a X 0 dz. Equation (37) implies that either r 0orr 0 meaning that the modes of the system may be unstable or stable respectively. Thus it is concluded that the modes may be oscillatory or non-oscillatory. Equation (38) implies that i may be either zero or non-zero which signifies that the modes may be nonoscillatory or oscillatory respectively. In the absence of solute concentration ie.. S 0 Eq. (38) gives Eq. (39). As the terms inside the bracket in Eq. (39) are positive. So i 0 which assures that the oscillatory modes are not allowed also confirms the validity of the PES in the absence of solute concentration gradient with the condition that B. Hence the oscillatory modes are dominant due to the presence of solute concentration gradients only. f g Ta I T p r i rb r i pei 4 B I3 r pei 4 i 0 B r i d f r e I 7 I9d I 4 r i 0 6. OVERSTABILITY CASE (39) Here the possibility whether the observed instability may actually be overstability has been examined. Rewriting Eq. (6) in the following form i f G xipe i x i q E x ip i f G i xxipxiqeqx xipe i f G x cos i x i q E x ip B i B i Rx QRx cos i i xi q Exip Sx i f G i B i i xip Exip QSx cos xipe xip i f G i B i xipxipe i xi q E 4 Q x xcos Qx xcos TA x cos x ipe x iqe xipexiqe xip 0 (40) 689

10 K. Kumar et al. / JAFM Vol. 0 No. pp. x-x 07. Equating the real imaginary parts of Eq. (40) gives 3 G L4 f x L 4 3 x f G L5 L6L L x G L7 L 6 x L Qx x cos Qx xcos f f 3 L L 3 4 3R x G L 4 Bx L8L Rx f L0L pqe 4 QRx cos B x L pqe 3 4 Sx G L 4 Bx L3L 4 Sx f L5 L6 pp EQSx 4 cos B x L7 pp E Qx x 4 3 cos G L8 L 9 x L Qx x cos f f L L 3 Qx xcos x pqee Tx A cos x L0L 0 (4) 3 G L4 f L L3 x 4 3 f G L5 L6 x L x 4 3 Qx xcos G L 7L 6 L L 3 Qx f f x L x 3 5 cos Rx G L 4 L0L pqe Rx f Bx L8L 9 QRx 3 5 cos L L3 L 4 Sx G L L L6 pp ESx 3 f Bx L5 L 6 L7 L 8 QSx 4 3 cos Qx xg L8L 9 LL cos Qx xcos f f 3 4 x L Qx xcos L TA xcos 30 L L 3 0 (4) where the symbols G L L3are defined as G D A G x P P L x pqe pq EE ppe L x p qe pe L3 ppqee G f f G pb xp qe x L4 G f L 5 L6 G f L7 G f L8 L9 x B p q E p x p q E p q E B L0 x x B B p qe p qeb x p qe pqeb x x p B p q E B L L L3 B x p qe 3 pb xp pe x x B p p E L4 p x p pe pp E B L5 x x B B p pe p x B p pe L6 pp EB x p pe ppeb x L7 B x p pe 4 3 L8 G f L9 G f L0 x p pq EE ppe pqe qe p x B L pp qee L B x x p q E B L3 x pqeb 3 L4 B pq E L 5 pb x p pe x x BppE 3 L6 p x p pe p pe B L7 B x pe p x B L8 x p pe pp E B 3p q L9 xqe pe L30 x E pe L3 x ppqee pp E qp E. (43) 690

11 K. Kumar et al. / JAFM Vol. 0 No. pp. x-x 07. Eliminating R from Eqs. (4) (4) assuming z an eight degree polynomial in z is obtained as z z z 3z 4z 5z 6z 7z 80 (44) where pp q EE x x (45) G x QBxcos x x B 8 G xq xcos xg 3 x 4 Qx cos Tx A xcos G xx BpE pg xx Bf G x x QB cos G pe pg x x QqE Bcos G Q x x Bf 6 5 cos G G S x x B pe qe GQS x x B cos pe qe TAQ x x Bcos pe pg x x QBcos pe pq x x G pe pbcos Q x x qe Bcos G GTA x x peb cos G Q x x peb cos QS x x pe qe B cos Q x x f Bcos Q x x Bcos pe p x 6 x f Bcos G Q TA x (46) The coefficients are quite large also of trivial importance in determining the overstability of the system. Since should be real for overstability to occur therefore all the roots of Eq. (44) should be positive. From Eq. (44) the product of roots 8 i.e. 0 positive this has to be negative for the nonhappening of overstability. Since 0 is always positive as obvious from Eq. (45) 8 will be negative if A. B G pe p G pe p pe qe G Q T x (47) Thus the aforementioned inequalities are the sufficient conditions for the non-occurrence of overstability the violation of which does not assure the possibility of overstable modes. 7. CONCLUSION The onset of double-diffusive convection in a couple-stress porous fluid layer with the presence of magnetic field rotation suspended particles was analyzed analytically using the linear stability theory. The Brinkman model is employed for the momentum equation. For the stationary state it is found that the stable solute gradient rotational parameter rule out the possibility of the onset of convection whereas suspended particles medium porosity accelerate the onset of convection. For a rotating medium the magnetic field medium permeability couplestress Darcy-Brinkman parameter have dual effects whereas for a non-rotating system the parameters like magnetic field couplestress Darcy-Brinkman have stabilizing effects medium permeability has a destabilizing effect these effects are also depicted both analytically graphically. The principle of exchange of stabilities is found to hold true in the absence of solute gradient parameter for B. Hence the oscillatory modes are dominant due to the presence of stable solute gradient parameter only. S Also the sufficient conditions for the nonhappening of overstability are obtained. REFERENCES Bansal J. L. (004). Viscous fluid dynamics. Oxford IBH Publishing Company Delhi India. Batchelor G. K. (967). An Introduction to Fluid Dynamics. Cambridge University Press London. Boussinesq J. (903). The analytical theory of heat Gauthier-Villars Paris 7. Brinkman H. C. (947a). A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. () Brinkman H. C. (947b). On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. () Chrasekhar S. C. (98). Hydrodynamic Hydromagnetic Stability Dover Publication New York. Drazin P. G. W. H. Reid (98). Hydrodynamic Stability. Cambridge University Press Cambridge. Gaikwad S. N. M. Dhanraj (06). Onset of Darcy-Brinkman reaction convection in an anisotropic porous layer. Journal of Applied Fluid Mechanics 9 ()

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